94-79 Gesztesy F., Simon B.

Rank One Perturbations at Infinite Coupling (17K, AMSTeX) Mar 31, 94

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Abstract. We discuss rank one perturbations $A_{\alpha}=A+\alpha (\varphi, \cdot)\varphi, \alpha\in\Bbb R, A\geq 0$ self-adjoint. Let $d\mu_{\alpha}(x)$ be the spectral measure defined by $(\varphi, (A_{\alpha}-z)^{-1}\varphi)=\int\, d\mu_{\alpha}(x)/(x-z)$. We prove there is a measure $d\rho_{\infty}$ which is the weak limit of $(1+\alpha^{2})\,d\mu_{\alpha}(x)$ as $\alpha\to\infty$. If $\varphi$ is cyclic for $A$, then $A_\infty$, the strong resolvent limit of $A_\alpha$, is unitarily equivalent to multiplication by $x$ on $L^{2}(\Bbb R, d\rho_{\infty})$. This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.

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